Aidy Boothroyd's side, who only needed a point, booked their spot in Italy and San Marino with a game to spare following the cruise in Chesterfield.
Ademola Lookman, Ezri Konsa and Calvert-Lewin scored first-half goals to put England in total control as debutant Phil Foden also impressed.
Everton's Calvert-Lewin added a fourth from the spot immediately after the break before substitutes Dominic Solanke and Reiss Nelson scored, with Christian Garcia's own goal wrapping it up late on.
The Young Lions were beaten on penalties in the semi-final by Germany at Euro 2017 but they have now reached seven straight tournaments, going 34 games unbeaten in qualification.
The Group Four winners face Scotland on Tuesday with Boothroyd expected to make several changes as his planning for Euro 2019 begins.
It took England, who had already gone close through Dael Fry, just nine minutes to get on the road to the Championship.
The marauding Jonjo Kenny crossed for Everton team-mate Lookman, who turned in the area and fired low past Francisco Pires from 10 yards.
From then it was a stroll, with Dean Henderson a spectator in goal, against a side who had scored just once in qualifying.
Foden shot straight at Pires on 22 minutes and the goalkeeper needed to make a smart save from the midfielder's teasing cross soon after.
He made another fine stop from Calvert-Lewin but the hosts doubled their lead when Brentford's Konsa drilled in after Pires parried Lewis Cook's long-range effort.
Winless Andorra never looked like troubling Henderson as they packed 10 men behind the ball to try to restrict the Young Lions.
But they failed to stop England adding a third before the break when Calvert-Lewin tapped in in stoppage time after Lookman beat Pires to the ball.
It then look just three minutes of the second half for the hosts to add a fourth when Calvert-Lewin emphatically scored from the spot after Konsa was tugged in the area.
England pressed for a fifth and Ryan Sessegnon's header was cleared off the line before the underemployed Henderson had to be alert to deny Alex Martinez a surprise consolation.
Tom Davies hooked over from six yards after Pires went AWOL with 25 minutes left and Foden's half volley was beaten away by Pires 10 minutes later.
Debutant Nelson, on as a substitute, also hit a post but Liverpool's Solanke did add a fifth with nine minutes remaining when he latched onto Foden's pass and rounded the goalkeeper to score.
Nelson then grabbed his first goal in stoppage time when his cheeky back-heel crept over the line and Garcia completed Andorra's wretched night when he turned past his own goalkeeper.